Control of Thermoacoustic Instabilities:Actuator-Sensor Placement

Pushkarini Agharkar, Priya Subramanian, Prof. R. I. SujithDepartment of Aerospace EngineeringProf. Niket KaisareDepartment of Chemical Engineering

Indian Institute of Technology, Madras

Acknowledgements: Boeing Travel Grant, IIT MadrasAlumni Affairs Association, IIT Madras

Thermoacoustic Instabilities

Occur due to positive feedback mechanism between combustion and acoustic subsystems

Representative system:

ducted premixed flame

Schuller (2003)

Acoustics

Heat Release

Modelof the ducted premixed flame

Control Framework • LQ Regulator

• Kalman filterActuator Placement

•LMI based techniques

•based on Hankel singular valuesConclusions

Model of the ducted premixed flame

• acoustic subsystem

• combustion subsystem

• single actuator and sensor pair

• actuator adds energy to the system

• sensor measures acoustic pressure

• Governing equation (linear)

Combustion Subsystem

1 2 11

cos ... discretized front tracking equationN

ia j i i

j

dHj y H H

dt

represents the monopole strength of

the flame element averaged over the

cross-sectional area of the duct.

i

th

H

i

Nonlinearity introduced due to

combustion subsystem

Acoustic Subsystem

• Governing equations:

1

...momentum

2 2 sin ...energy

2

j

P

j j f ii

dj

dt j

dj j j y H

dt j

sincaj y u

M

contribution from controller

fluctuating heat release

1

1

acoustic velocity cos

acoustic pressure sin

N

jj

N

jj

u j y t

Mp j y t

j

Properties of the Model

– Non-normality: due to coupling between combustion and acoustic subsystems

– Nonlinearity: due to the equations of evolution of the flame front

– Motivation: Reducing the transient growth and avoiding triggering

State-Space Representation

1

1 2 11

2 2 2 sin sin

cos

j

Pc

j j f i ai

Ni

a j i ij

dj

dt j

dj j j y H j y u

dt j M

dHj y H H

dt

=d

dt

A B u

u K

2

2 2

1 1

N Pj

j ij i

T

jJ t H t

j

is minimized.

Tcl u u

such that the cost functional

Linear Quadratic (LQ) Regulator

Linear Quadratic (LQ) Regulator

u K

Open loop plant :(without control)

dA

dt

dA Bu

dtA BK

Closed loop plant :

(with control)

cA

LMI optimization problem

dA Bu

dtA BK

variables: , cP

is the upper bound on the

energy of the plant

controlled using the LQ

Regulator

c

T

min :

0 0

c

H Hc c

c

PA A P P P

I P I

- Linear Matrix Inequalities (LMI): inequalities defined for matrix variables

cA

Actuator Placement using LMI based Optimization Techniques

10cl

max

* Actuator located closer to flame

results in lower bounds on c E

* This feature is highlighted when

the controller is aggressive ( 1)cl

1cl

Controllability–Observability Measures

• Other ways to determine optimal placement of actuators and sensors

• Controllability-Observability measure based on Hankel singular values (HSVs).

– measure =

– Hankel singular value

2

ii

Controllability–Observability Measures• Measure of controllability-

observability based on HSVs calculated for various actuator and sensor locations

• Locations of the antinodes of the third acoustic pressure mode give highest measure

• From numerical simulations, the third acoustic mode is also the highest energy state

The techniques give contradictory results

Antinodes of the least

stable modes

Measures based on HSVs.

Locations closer to the

flame

LMI based techniques

Actuator Placement Numerical Validation

open loop

0.3

0.5a

a

y

y

In the presence of transient growth, actuators placed according to LMI techniques give better performance than when placed based on HSV measures

Actuator Placement Numerical Validation

0.3

0.5

0.833

In the absence of transient growth, actuators placed according to HSV measures give better performance than in the presence of transient growth, but still not better than LMI techniques.

Conclusions

• Actuator-Sensor placement of non-normal systems requires different approaches than the ones used conventionally.

• For the ducted premixed flame model, actuators placed nearer to the flame give better overall performance.

• Controllers based on these actuators results in low transient growth as well as less settling time.